How are new entries being made for requests? Would you only want to calculate the Approved amounts? Are you wanting this to be on each row, on a single row for each student, or on a separate sheet entirely?

Having the separate sheet for the starting balance would certainly be very helpful. once that is set up...

[Annual Funding Max]:

=INDEX({Other Sheet Max Amount}, MATCH({Other Sheet Employee ID}, [Employee ID#]@row, 0))

[Annual Funding Remaining]:

=[Annual Funding Max]@row - SUMIFS([Approved Reimbursement Amount:]:[Approved Reimbursement Amount:], [Employee ID#]:[Employee ID#], [Employee ID#]@row)

Do any of your employee ID's start with a leading zero?

We can get a running total that only looks at previous entries with the help of a system generated Created (date) column.

[Annual Funding Remaining]:

=[Annual Funding Max]@row - SUMIFS([Approved Reimbursement Amount:]:[Approved Reimbursement Amount:], [Employee ID#]:[Employee ID#], [Employee ID#]@row, Created:Created, @cell <= Created@row)

Here is why I asked about leading zeros... If you have a leading zero then the data is a text value. If you do not have a leading zero then the data is stored as a numerical value. Inconsistent data types will almost always cause headaches.

Insert a helper column (that can be hidden to keep the sheet looking clean) and use this column formula:

[Employee ID#]@row + ""

The + "" will convert everything including the numerical values into text strings across the board. Now you have a column that has a consistent data type throughout. Reference this column in all of your formulas instead of the original column with mixed data types.